Of course there are other ways to show a function is increasing (the same argument can be applied to decreasing) on an interval. For example, using the definition of increasing: if x,y are in [a,b] then show y > x implies f(y) >= f(x).

Example: Show that f(x) = x^2 is increasing on [0,2].
Let x and y belong to [0,2] with y > x. Then we can write y = x + e, for some e>0. Then f(y) = (x+e)^2 = x^2 + 2xe + e^2 > f(x) = x^2, since e^2 is >0 and 2xe is >0.